Two polynoms

[YuK]

For what degrees N are there two polynoms of degree N with the integer coefficients, each has N various integer roots, and polynoms difference - constant? (problem from D. G. Fon-Der-Flaas).

Example for N=3

p(x)=x*(x-1)*(x-8) and q(x)=(x-5)*(x-6)*(x+2)

p(x)-q(x)= - 60

[YuK]

Example for N=5.


YuK

[AlvaroDíaz]

Hi Yuk. Time to post more tips, isn't?

Regards. Alvaro.

[YuK]

Hi Alvaro.

Time to post more tips. Peoples knows solution for N=6,7,8,9,(11?). But I dont find this solutions.

Regards
Yuk

[ptc-1368288]

On 11/21/2009 10:34:01 PM, YuK wrote:
>Hi Alvaro.
>
>Time to post more tips.
>Peoples knows solution for
>N=6,7,8,9,(11?). But I don't
>find this solutions.
>
>Regards
>Yuk
_____________________________

There may be connection with "cyclotomic", but it does look so.
Maybe with partitioning ?

jmG

[YuK]

N=6. I find two polynoms for N=6 with roots (1,-1,10,-10,9,-9), (5,-5,6,-6,11,-11).

YuK

[YuK]

I see the symmetry of polinom roots for solution. And I try all symmetry variant roots for N=7 with integers abs(x_i)<56, (y_i)<56, but I don't find here the solution.

P1(x)= (x-x1)*(x-x2)...(x-x7)
P2(y)= (y-y1)*(y-y2)...(y-y7)

x1 < y1 < y2 < x2 < x3 < y3 < y4 < x4 < x5 < y5 < y6 < x6 < x7 < y7,

x1-y1=x7-y7,
y1-y2=x6-x7,
y2-x2=y6-x6,
x2-x3=y5-y6 etc.

Control equations

x1+x2+x3+..+x7=y1+y2+y3+..+y7

x1*x2+x1*x3+..+x6*x7=y1*y2+y1*y3+..+y6*y7

x1*x2*x3+x1*x3*x4+..+x5*x6*x7=y1*y2*y3+y1*y3*y4+..+y5*y6*y7 etc.

YuK

[YuK]

Example for N=7. It is the rezult of my relax NY time.